For \(0\leqslant \delta \leqslant 1-q^{-1}\) , the asymptotic Gilbert-Varshamov bound of locally repairable codes is given in [1]} by
\(R_q(r,\delta )\geqslant 1-\min _{0<s\leqslant 1} \Big {\lbrace }\frac{1}{r+1} \log _q\Big ([1+(q-1)s]^{r+1}+(q-1)(1-s)^{r+1}\Big )-\delta \log _qs\Big {\rbrace }.\) 
